Whakaoti i te 2x^2+6x^2+7x+60=0

$$2 { x }^{ 2 } +6 { x }^{ 2 } +7x+60 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-7+\sqrt{1871}i}{16}\approx -0.4375+2.703441094i$$

$$x=\frac{-\sqrt{1871}i-7}{16}\approx -0.4375-2.703441094i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te 2x^{2} me 6x^{2}, ka 8x^{2}.

$$8x^{2}+7x+60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 7 mō b, me 60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 8\times 60}}{2\times 8}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 8\times 60}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-7±\sqrt{49-32\times 60}}{2\times 8}$$

Whakareatia -32 ki te 60.

$$x=\frac{-7±\sqrt{49-1920}}{2\times 8}$$

Tāpiri 49 ki te -1920.

$$x=\frac{-7±\sqrt{-1871}}{2\times 8}$$

Tuhia te pūtakerua o te -1871.

$$x=\frac{-7±\sqrt{1871}i}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-7±\sqrt{1871}i}{16}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{1871}i}{16} ina he tāpiri te ±. Tāpiri -7 ki te i\sqrt{1871}.

$$x=\frac{-7+\sqrt{1871}i}{16}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{1871}i}{16} ina he tango te ±. Tango i\sqrt{1871} mai i -7.

$$x=\frac{-\sqrt{1871}i-7}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{-7+\sqrt{1871}i}{16} x=\frac{-\sqrt{1871}i-7}{16}$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te 2x^{2} me 6x^{2}, ka 8x^{2}.

$$8x^{2}+7x+60=0$$

Tangohia te 60 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$8x^{2}+7x=-60$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+7x}{8}=-\frac{60}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\frac{7}{8}x=-\frac{60}{8}$$

Whakahekea te hautanga \frac{-60}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}+\frac{7}{8}x=-\frac{15}{2}$$

Whakawehea te \frac{7}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{16}. Nā, tāpiria te pūrua o te \frac{7}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{8}x+\left(\frac{7}{16}\right)^{2}=-\frac{15}{2}+\left(\frac{7}{16}\right)^{2}$$

Pūruatia \frac{7}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{8}x+\frac{49}{256}=-\frac{15}{2}+\frac{49}{256}$$

Tāpiri -\frac{15}{2} ki te \frac{49}{256} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{7}{8}x+\frac{49}{256}=-\frac{1871}{256}$$

Tauwehea x^{2}+\frac{7}{8}x+\frac{49}{256}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{16}\right)^{2}=-\frac{1871}{256}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+\frac{7}{16}\right)^{2}}=\sqrt{-\frac{1871}{256}}$$

Whakarūnātia.

$$x+\frac{7}{16}=\frac{\sqrt{1871}i}{16} x+\frac{7}{16}=-\frac{\sqrt{1871}i}{16}$$

Me tango \frac{7}{16} mai i ngā taha e rua o te whārite.

$$x=\frac{-7+\sqrt{1871}i}{16} x=\frac{-\sqrt{1871}i-7}{16}$$